Problem 30 - Algebra 2

Question

Write an equation for each conic section. Then sketch the graph. ellipse with center \((3,2),\) vertices \((9,2)\) and \((-3,2),\) and co-vertices \((3,5)\) and \((3,-1)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{(x-3)^2}{36}+\frac{(y-2)^2}{9} = 1\). The sketch would show an ellipse centered at point \((3,2)\) with its major axis along the x-axis and measures \(12\) units and the minor axis measures \(6\) units along the y-axis.

Step 1: Calculate the lengths of the semi-major and semi-minor axes

In our given ellipse, the center is \((h,k) = (3,2)\). The vertices are at \((9,2)\) and \((-3,2)\), so the distance from the center to each vertex along the x-axis is \(a = 9 - 3 = 6\) and \(-3 - 3 = -6\). So, \(a = 6\). The co-vertices are at \((3,5)\) and \((3,-1)\), so the distance from the center to each co-vertex along the y-axis is \(b = 5 - 2 = 3\) and \(-1 - 2 = -3\). So, \(b = 3\).

Step 2: Write the equation of the ellipse

The standard form for the equation of an ellipse with a horizontal major axis is \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\). Substituting for \(h, k, a\), and \(b\), we get the equation as \(\frac{(x-3)^2}{6^2}+\frac{(y-2)^2}{3^2} = 1\). This simplifies to \(\frac{(x-3)^2}{36}+\frac{(y-2)^2}{9} = 1\)

Step 3: Sketch the graph of the ellipse

Using the center, vertices, and co-vertices, a sketch of an ellipse can be created. The center is at \((3,2)\), the endpoints of the horizontal axis (major-axis) are at \((9,2)\) and \((-3,2)\), and the endpoints of the vertical axis (minor-axis) are at \((3,5)\) and \((3,-1)\). Draw an ellipse that encapsulates these points, marking the important points discussed.

Key Concepts

Conic Sections

The study of conic sections is important in understanding curves formed by the intersection of a plane and a double-napped cone. These sections include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has a unique, standard equation, which helps us identify and differentiate them easily.

When a plane cuts through a cone parallel to its base, the resulting shape is an ellipse. Unlike a circle, an ellipse has two axes: the major axis, which is the longest diameter, and the minor axis, which is the shortest. This unique shape is what makes ellipses interesting and often a focus of study in mathematics.

• Circle: A special type of ellipse where the two axes are equal.
• Ellipse: Defined primarily by its center, major and minor axes.

Understanding conic sections can open doors to exploring more complex polynomial equations and can be applied in various fields such as astronomy, physics, and engineering.

Ellipse Properties

An ellipse is defined by several key properties, which differentiate it from other geometric shapes. The principal properties include its center, vertices, co-vertices, and axes.

An ellipse's center is the midpoint of both its major and minor axes. For the ellipse given in the problem, the center is at ackslash((3,2)ackslash). The major axis is the longest line that can pass through the center, stretching across the ellipse from one vertex to its opposite vertex.

- **Vertices**: Points on an ellipse that are furthest from the center along the major axis. In the exercise, these are ackslash((9,2)ackslash) and ackslash((-3,2)ackslash).
- **Co-vertices**: Points along the minor axis that are closest to the center. Here, they are ackslash((3,5)ackslash) and ackslash((3,-1)ackslash).
- **Axes Lengths**: The distance from the center to a vertex is known as the semi-major axis, denoted by ackslash(aackslash). Similarly, the distance from the center to a co-vertex is called the semi-minor axis ackslash(backslash). In this ellipse, ackslash(a=6ackslash) and ackslash(b=3ackslash).

These properties are essential in forming the standard equation of an ellipse, which helps graph its precise shape.

Graphing Ellipses

Graphing an ellipse involves plotting its defining components such as the center, vertices, and co-vertices.

Start by identifying the center of the ellipse. In this case, it's ackslash((3,2)ackslash). From the center, measure along the x-axis to plot the vertices ackslash((9,2)ackslash) and ackslash((-3,2)ackslash), and along the y-axis for the co-vertices ackslash((3,5)ackslash) and ackslash((3,-1)ackslash). These points give a framework to draw the ellipse.

Next, draw an elongated, smooth curve that touches all these significant points. The curve must reflect the largest span along the major axis, while the minor axis determines the narrowness of the ellipse. You're basically outlining an oval that connects the plotted points symmetrically.

• **Major Axis:** The longest diameter across the ellipse, connecting the vertices.
• **Minor Axis:** The shortest diameter, perpendicular to the major axis.

Understanding how to accurately graph an ellipse enhances your comprehension of its geometry and construction, allowing for better visualization of this key conic section.